Definition:Bounded Above Mapping/Real-Valued

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This page is about real-valued functions which are bounded above. For other uses, see Definition:Bounded Above.


Let $f: S \to \R$ be a real-valued function.

$f$ is bounded above on $S$ by the upper bound $H$ if and only if:

$\forall x \in S: \map f x \le H$

That is, if and only if the set $\set {\map f x: x \in S}$ is bounded above in $\R$ by $H$.

Unbounded Above

Let $f: S \to \R$ be a real-valued function.

Then $f$ is unbounded above on $S$ if and only if it is not bounded above on $S$:

$\neg \exists H \in \R: \forall x \in S: f \left({x}\right) \le H$

Also see